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Advances in Optics and Photonics


  • Editor: Bahaa E. A. Saleh
  • Vol. 2, Iss. 4 — Dec. 31, 2010

Two-photon absorption: an overview of measurements and principles

Mariacristina Rumi and Joseph W. Perry  »View Author Affiliations

Advances in Optics and Photonics, Vol. 2, Issue 4, pp. 451-518 (2010)

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The range of organic compounds whose degenerate two-photon absorption (2PA) spectrum has been reported has increased rapidly in recent years, in parallel with the growing interest in applications based on the 2PA process. The comparison of results from different techniques is not always straightforward, and experimental conditions employed may vary significantly. We overview the concepts underlying 2PA measurements and the common assumptions and approximations used in the data analysis for various techniques. The importance of selecting appropriate excitation regimes under which measurements should be performed and of avoiding contributions from absorption mechanisms in addition to 2PA will be emphasized.

© 2010 Optical Society of America

ToC Category:
Nonlinear Optics

Original Manuscript: March 26, 2010
Revised Manuscript: July 7, 2010
Manuscript Accepted: July 8, 2010
Published: August 26, 2010

Virtual Issues
(2010) Advances in Optics and Photonics

Mariacristina Rumi and Joseph W. Perry, "Two-photon absorption: an overview of measurements and principles," Adv. Opt. Photon. 2, 451-518 (2010)

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  104. For example, in some cases [101, 102, 103], the cross section is defined to “count” directly the number of molecules excited, instead of the photons absorbed, leading to the relationship nm(2)=δ̃Ngϕ2.(I)    In this case, the propagation equation (1) becomesdϕdz=−2δ̃Ngϕ2.(II)    Obviously, the right-hand sides of the two versions of the propagation equation [Eqs. (1, II)] and of the equation describing the excitation rate due to 2PA [Eqs. (6, I)] differ only by a constant factor and become equivalent if δ=2δ̃.(III)
  105. More precisely, any absorption process that would take place at this stage, via 1PA, 2PA or other nonlinear processes, would reflect properties of the excited state r, not of the ground state g, and thus describe a very different sample, from a spectroscopic point of view. All processes, though, contribute to the attenuation of the propagating beam and Eq. (1) would have to be appropriately modified. This situation will also be discussed in Subsection 3.2. ESA following 2PA will also be considered in Subsection 4.3.
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  107. It is generally assumed that the fluorescence quantum yield is the same after 1PA and 2PA excitation, as for most molecules the system relaxes, after absorption, to the lowest excited state, r, by internal conversion, irrespective of the photon energy and the manner of excitation, as discussed above. However, there are exceptions to this general rule, as some molecules are known that exhibit a wavelength dependent quantum yield or emission spectrum, and fluorescence emission from upper excited states.
  108. Δm(2) is equivalent to the “saturation parameter” α discussed by Xu and Webb [17], pp. 518–520.
  109. It should be kept in mind that Eq. (14) was obtained under a series of assumptions (for example regarding the change in photon flux through the sample). Thus, strictly speaking, Eq. (14) is also an approximate description of the fraction of molecules excited.
  110. The total number of photons absorbed per pulse depends on the excitation volume. For a collimated beam, as the case considered in this section, the volume scales with the sample path length L. However, for a focused beam, only a path length of the order of twice the Rayleigh range of the beam, z0=πw02n∕λ (if this is shorter than L), may have to be considered [29], at least as a first approximation, because of the decrease in photon flux away from the focal plane. See also examples in Subsection 3.1b. We have used here for the Rayleigh range z0 the definition typically introduced for Gaussian beams, for which it corresponds to the distance along z between the waist and the point at which the beam radius is 2w0 [111]. The term confocal parameter is also sometimes used when referring to the quantity z0 (see, for example, [112], p. 117) or 2z0 [see, for example, H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966)]. [CrossRef] [PubMed]
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  114. The 1PA equivalent of parameter Δph(2) is σN0L, the fraction of light absorbed in the limit of thin sample or weak absorption.
  115. In the case of the 2PIF method, an additional limitation on the concentration is imposed by reabsorption of the emitted photons within the material (inner filter effect). This effect is observed to different extents depending on how the fluorescence signal is collected (for example, under backscattering conditions or at 90° with respect to the incident beam).
  116. J. Segal, Z. Kotler, M. Sigalov, A. Ben-Asuly, V. Khodorkovsky, “Two-photon absorption properties of (N-carbazolyl)-stilbenes,” Proc. SPIE 3796, 153–159 (1999). [CrossRef]
  117. D. Beljonne, W. Wenseleers, E. Zojer, Z. Shuai, H. Vogel, S. J. K. Pond, J. W. Perry, S. R. Marder, J.-L. Brédas, “Role of dimensionality on the two-photon absorption response of conjugated molecules: the case of octupolar compounds,” Adv. Funct. Mater. 12, 631–641 (2002). [CrossRef]
  118. B. Strehmel, A. M. Sarker, H. Detert, “The influence of σ and π acceptors on two-photon absorption and solvatochromism of dipolar and quadrupolar unsaturated organic compounds,” ChemPhysChem 4, 249–259 (2003). [CrossRef] [PubMed]
  119. G. P. Bartholomew, M. Rumi, S. J. K. Pond, J. W. Perry, S. Tretiak, G. C. Bazan, “Two-photon absorption in three-dimensional chromophores based on [2.2]-paracyclophane,” J. Am. Chem. Soc. 126, 11529–11542 (2004). [CrossRef] [PubMed]
  120. S. K. Lee, W. J. Yang, J. J. Choi, C. H. Kim, S.-J. Jeon, B. R. Cho, “2,6-Bis[4-(p-dihexylaminostyryl)-styryl]anthracene derivatives with large two-photon cross sections,” Org. Lett. 7, 323–326 (2005). [CrossRef] [PubMed]
  121. When a photon counting approach is used, signal discrimination procedures can be used to reject noise.
  122. R. L. Swofford, W. M. McClain, “The effect of spatial and temporal laser beam characteristics on two-photon absorption,” Chem. Phys. Lett. 34, 455–460 (1975). [CrossRef]
  123. We will introduce beams with a Gaussian spatial profile in Subsection 3.2. To explain the results in Fig. 4, it is sufficient to mention that the beam size depends on z as follows: w0(1+(z∕z0)2)1∕2. The total number of excited molecules is obtained by integrating Nm(2)(τ) over the excitation volume. The result for a generic path length L and for a sample with the focal plane at L∕2 (and with photon flux ϕ0 at z=0) is ∫VNm(2)(τ)dV=12δτN0∫−L∕2L∕2dz∫0∞2πrdrϕ2=12δτN0πw02z0ϕ022arctanL2z0=δN0Nph22τ(2nλarctanL2z0).   The asymptotic value for L≫z0 is in this case δN0Nph2πn∕(2τλ).
  124. M. Rumi, J. E. Ehrlich, A. A. Heikal, J. W. Perry, S. Barlow, Z. Hu, D. McCord-Maughon, T. C. Parker, H. Röckel, S. Thayumanavan, S. R. Marder, D. Beljonne, J.-L. Brédas, “Structure-property relationships for two-photon absorbing chromophores: bis-donor diphenylpolyene and bis(styryl)benzene derivatives,” J. Am. Chem. Soc. 122, 9500–9510 (2000). [CrossRef]
  125. For example, if the flux is reduced by 5% by 2PA [ϕ(z)∕ϕ(0)=0.95] and each time the photon flux is measured with an uncertainty of 2% (the two measurements before and after the beam are independent), from error propagation, the relative uncertainty in δ would be over 50%.
  126. It should be kept in mind that this common definition of β implicitly assumes that approximation (ii) is valid, as this parameter is proportional to N0. β, usually called the “2PA coefficient,” is a macroscopic equivalent of the molecular parameter δ. In cases for which (ii) is not valid, β is not a constant of the material, but depends on ϕ through Ng, the concentration of molecules in the ground state (Ng≤N0).
  127. V. I. Bredikhin, M. D. Galanin, V. N. Genkin, “Two-photon absorption and spectroscopy,” Sov. Phys. Usp. 16, 299–321 (1974). [CrossRef]
  128. In Subsection 4.3 we will discuss how ESA can also limit the applicability of Eq. (29) in NLT measurements, especially for ns pulse durations. At the moment, we are considering instead the case of materials for which 2PA is the only absorption process that can take place at the excitation wavelength λ.
  129. In this context, the first argument for the function ϕ is the coordinate along the propagation direction, z, the origin being the front face of the material; the second argument refers to the radial distance, r, from the axis z. Thus, in this notation ϕ(0,0) corresponds to the peak on-axis flux at the entrance of the material.
  130. The initial pulse energy is now E(0)=τEph∫0∞ϕ(0,0)e−2r2∕w022πrdr=τEphϕ(0,0)πw02∕2.
  131. When the results for the f/60 case in Fig. 7 are compared with those in Fig. 6 for the same pulse energy, it can be seen that the transmittance is larger in the former situation, even when ground state depletion is neglected. This is due to the effect of focusing: even if L<z0 in the f/60 case, the beam size increases slightly away from the focal plane within the sample; instead, the beam was assumed to be perfectly collimated in Fig. 6.
  132. This change in beam size is indeed exploited in closed-aperture Z-scan experiments [30] to measure the nonlinear refractive index of a material (which is related to the real part of the susceptibility χ(3)).
  133. To differentiate between the examples discussed above, where z represented the position within the sample along the propagation direction, and the current example, where z is the position of the front face of the sample with respect to the focal plane, the z dependence of ϕ is now expressed with a subscript.
  134. This is true not only for a Gaussian beam, but also for a beam with a generic beam profile, as long as the z dependence of the flux is known.
  135. For this type of beam the relationship between pulse energy and peak flux isE=Eph∫0∞ϕ0(r=0)e−2r2∕w022πrdr∫−∞∞e−t2∕τ̃2dt=Ephϕ0(r=0)ππw02τ̃2.
  136. The value of the transmittance at a generic point within the sample can be obtained by substituting z′, the distance of the point from the front face, for L.
  137. It should be remembered that we are assuming that 1PA absorption is negligible.
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  147. In this context, we define the time average of a function Y as follows:⟨Y(t)⟩=f∫TT+1∕fY(t)dt,   where T is a generic time and Y(t) is a periodic function with period 1∕f.
  148. This assumes that τfl≪1∕f, so that effectively all molecules that have been excited during the pulse duration τ have decayed back to the ground state. For other indirect methods 1∕f needs to be large with respect to the time constant of the process to be monitored (thermal time constant, phosphorescence lifetime,…). If this requirement is not satisfied, the sample conditions are different every time a new pulse arrives, and ground state depletion may become relevant after a large number of pulses. In the case of single pulse measurements, the integration time for the signal needs to be long with respect to the time constant of the process monitored, if results for different materials are to be compared under the same excitation conditions.
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  152. For ideal monochromatic light with constant intensity and in the absence of noise, g(2)=1. For a square pulse of duration τ, g(2)=(τf)−1. For a Gaussian pulse with 1∕e width τ, g(2)=(τf(2π)1∕2)−1.
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  167. An expression similar to Eq. (46) could be written in the case of a molecule undergoing intersystem crossing from r to the lowest level in the triplet manifold and then being excited to a higher-lying triplet state. In this case, σex would represent a triplet–triplet ESA cross section.
  168. The quantity ϕτ corresponds to the photon fluence, or the number of photons per unit area of the beam, Nph∕a [Nph from Eq. (17) for the case of a pulse with constant profile in space and time].
  169. Even if ESA occurs from the triplet state, as described in note [167], the fluorescence quantum yield and intersystem crossing rates would be unchanged if Nex is small, and thus ⟨S⟩ would still not be affected by ESA and would not depend on pulse duration.
  170. This numerical example referred to excitation with ns pulses. However, similar conditions for ESA can be reached by using fs pulses from an amplified laser, for example, with ϕ values only slightly larger than those considered in case 3.1.b.
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  173. Unfortunately, there are some typographical errors in the solution of the system of equation reported by Kleinschmidt et al. [162].
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  188. The case of different excitation volumes in the two samples could be handled by including appropriate correction terms in the collection efficiency factor, G. Beside the obvious case of samples with different thicknesses in a collimated beam, the excitation volume is also different if the samples do not have the same refractive index and L≫z0.
  189. Due to the relatively low concentration used in 2PA measurements, it is often assumed that the refractive index of the solution is the same as that of the pure solvent.
  190. For example, the refractive index dependence of G varies with the optical setup configuration on both the excitation and the emission sides. A few cases, often used in the literature, are briefly considered here. If the excitation beam is collimated and the fluorescence signal is imaged at 90° with respect to the excitation propagation direction [124], the situation is similar to that employed in many commercial spectrofluorimeters, for which the signal dependence on nfl is described as nfl−2 [113]. If the excitation beam is focused in the sample, the excitation volume in the material depends on n, and consequently G depends on both n and nfl. In the focused beam configuration described by Beljonne, et al. M. A. Albota, C. Xu, W. W. Webb, “Two-photon fluorescence excitation cross sections of biomolecular probes from 690 to 960 nm,” Appl. Opt. 37, 7352–7356 (1998)]. [CrossRef]
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